Maintain Jump Combo Counter

In this game, if you chain together jumps - which can be on boxes or enemies - and certain jumps are on enemies, you can get special rewards. For example, your 5th jump, if it's on an enemy, will give you an extra life. However, you can maintain your invisible jump combo counter even if you hit the ground - the combo only resets if Crash is in his walking/standing animations. Thus, if you spin just before you hit the ground, and jump before you stop spinning, you can maintain your counter. This can useful in particular if you try to chain 200+ jumps together to get Crystals and other items (there's an guide in the FAQs with more in-depth details on this item glitch).

Start With All Power-Ups

After booting the game, wait at the title screen for the demo to start. Shortly before the game shows the demo of Bye Bye Blimps, begin holding Triangle. After the demo begins you will be in control of Coco. From here, pause and exit the level, and you will appear in the middle of the warp room (as if you had began a new game) with all the power ups.
Note: If you finish Bye Bye Blimps, you will be ejected in the fourth warp room, but the force field in front of the hallway will still be in place, trapping you there.

Fireworks Show and 45th Gem

Secrets

Hidden Level Eggipus Rex

To access the hidden level, Eggipus Rex, go to Dino Might and take the Yellow Gem Path to the secret area. While being chased, let the second pterodactyl attack you. Instead of killing you, it will carry you off to this secret level.